hw2 - Homework Set No 2 Physics 880.08 Deadline –...

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Unformatted text preview: Homework Set No. 2, Physics 880.08 Deadline – Wednesday, November 3, 2010 1. (10 pts) In class we showed that Dirac spinors transform as ψ D ( x ) → ψ ′ D ( x ′ ) = parenleftBigg e − i 2 vectorσ · ( vector θ + i vector ξ ) e − i 2 vectorσ · ( vector θ − i vector ξ ) parenrightBigg ψ D ( x ) (1) under Lorentz transformations. Show that this transformation rule is equivalent to ψ D ( x ) → ψ ′ D ( x ′ ) = e − i 4 ω μ ν σ μ ν ψ D ( x ) where σ μ ν = i 2 [ γ μ , γ ν ] . As usual ξ i = ω i and θ i = 1 2 ǫ ijk ω jk . You may consider boosts and rotations separately for the full credit. 2. (a) (5 pts) Complete the proof started in class that ¯ ψ γ μ ψ is a 4-vector by showing that it transforms like one under infinitesimal boosts. ψ is the Dirac spinor which transforms according to Eq. (1). (b) (5 pts) Prove that ¯ ψ γ μ γ 5 ψ is a 4-vector under both boosts and rotations. What happens to it under parity?...
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hw2 - Homework Set No 2 Physics 880.08 Deadline –...

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